一种基于同调扰动技术的通用凸惩罚迭代正则化方法用于Banach空间中的非线性反问题

Convergence Rates for Oversmoothing Banach Space Regularization

This paper studies Tikhonov regularization for finitely smoothing operators in Banach spaces when the penalization enforces too much smoothness in the sense that the penalty term is not finite at the true solution. In a Hilbert space setting, Natterer (1984) showed with the help of spectral theory that optimal rates can be achieved in this situation. ('Oversmoothing does not harm.') For oversmoothing variational regularization in Banach spaces only very recently progress has been achieved in several papers on different settings, all of which construct families of smooth approximations to the t…

Banach空间中Tikhonov型凸正则化 ill-posed线性方程收敛速率的反向结果

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

研究Banach空间中Tikhonov型正则化Ill-posed线性算子方程的收敛速率，通过变分源条件得到误差测度的上界，证明在扭曲Bregman距离情况下上界最优，扩展了从Hilbert空间到Banach空间的反向结果。

基于惩罚的凸变分正则化中的平滑性条件

Penalty-based smoothness conditions in convex variational regularization

本文研究了在Banach空间中定义的一般凸惩罚下的线性病态问题的Tikhonov正则化，通过引入仅涉及噪声自由最小化序列和未知惩罚最小化解的不等式来替代传统的平滑性假设，从而推导出Bregman距离的误差界，并展示了变分不等式如何保证这些假设的成立。

Banach空间中的Landweber-Kaczmarz方法与不精确内求解器

Landweber-Kaczmarz method in Banach spaces with inexact inner solvers

本文提出在Banach空间中使用不精确内求解器的Landweber-Kaczmarz方法，通过ε-次微分理论分析其收敛性，并结合Nesterov策略提出加速版本，通过计算断层成像和偏微分方程参数识别的数值实验验证方法有效性。

Tikhonov正则化与l^0项补充凸惩罚：在稀疏约束下的l^1收敛

Tikhonov regularization with l^0-term complementing a convex penalty: l^1 convergence under sparsity constraints

本文研究了在稀疏假设下，通过l^0正则化与一般凸稳定功能补充的Tikhonov正则化方法，分析了在l^1范数下误差估计及线性和亚线性收敛率的推导。

Bregman迭代变分正则化在反问题中的高阶收敛速率研究

Higher order convergence rates for Bregman iterated variational regularization of inverse problems

本文研究了在Banach空间中线性病态算子方程变分正则化解的收敛性，提出高阶变分源条件(VSCs)以获得最优收敛速率，并通过数值实验验证了理论结果。

序列空间尺度中带过平滑惩罚项的稀疏变分正则化

Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces

本文研究序列空间尺度上的线性不适定问题，分析 $\ell_p$ 与计数测度 $\ell_0$ 惩罚下的稀疏变分正则化，给出稳定性、先验收敛速率及过平滑情形的硬阈值分析方法。

具有一般残差项的正则化中源条件的新方法

A new approach to source conditions in regularization with general residual term

本文针对Banach或更一般的拓扑空间中非线性不适定算子方程，提出了一种结合近似源条件和变分不等式优势的收敛速率证明方法，能够处理一般非凸残差项和非光滑算子。

通过凸约束的两点梯度方法正则化反问题

Regularization of inverse problems by two-point gradient methods with convex constraints

本文提出并分析了一种基于Landweber迭代和外推策略的两点梯度方法，用于求解Banach空间中的反问题，能够利用非光滑惩罚项（如L^1范数和总变分型惩罚函数）来正则化解的稀疏性和分段常数性等特性。

On Tikhonov functionals penalized by Bregman distances

We investigate Tikhonov regularization methods for nonlinear ill-posed problems in Banach spaces, where the penalty term is described by Bregman distances. We prove convergence and stability results. Moreover, using appropriate source conditions, we are able to derive rates of convergence in terms of Bregman distances. We also analyze an iterated Tikhonov method for nonlinear problems, where the penalization is given by an appropriate convex functional.

Convergence rates of a dual gradient method for constrained linear ill-posed problems

In this paper we consider a dual gradient method for solving linear ill-posed problems $Ax = y$, where $A : X \to Y$ is a bounded linear operator from a Banach space $X$ to a Hilbert space $Y$. A strongly convex penalty function is used in the method to select a solution with desired feature. Under variational source conditions on the sought solution, convergence rates are derived when the method is terminated by either an {\it a priori} stopping rule or the discrepancy principle. We also consider an acceleration of the method as well as its various applications.

Maximal Spaces for Approximation Rates in $\ell^1$-regularization

We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $\ell^1$-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $L^2$-space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $\ell^2$-norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method al…

带有稀疏约束的Tikhonov函数全局最小化算法

A global minimization algorithm for Tikhonov functionals with sparsity constraints

本文提出了一种全局收敛算法，用于求解Banach空间中非线性正向算子的Tikhonov函数最小化问题，结合稀疏性促进惩罚项，并通过数值实验验证了理论结果。

使用多次重复测量数据求解不适定问题的对偶梯度方法

Dual gradient method for ill-posed problems using multiple repeated measurement data

针对线性不适定问题，利用多次重复独立同分布无偏测量数据的平均值，提出一种对偶梯度方法重构强凸罚函数下的最小化解，并通过先验停止规则或偏差原理的统计变体证明了收敛性及收敛速率。